Polar Coordinates

• May 6th 2010, 08:30 AM
Ricko
Polar Coordinates
Let $\displaystyle (x,y)$ be the common coordinates of $\displaystyle R^{2}$ and $\displaystyle V=R^{2}-\{(x,0):x\le 0\}$. On $\displaystyle V$ the polar coordinates are defined unambiguously for $\displaystyle 0<\theta<2\pi$. Find $\displaystyle \frac{\partial}{\partial{r}}$,$\displaystyle \frac{\partial}{\partial{\theta}}$. Find the relationship between these vectors and $\displaystyle \frac{\partial}{\partial{x}}|$,$\displaystyle \frac{\partial}{\partial{y}}|$.
• May 7th 2010, 02:43 AM
HallsofIvy
Quote:

Originally Posted by Ricko
Let $\displaystyle (x,y)$ be the common coordinates of $\displaystyle R^{2}$ and $\displaystyle V=R^{2}-\{(x,0):x\le 0\}$. On $\displaystyle V$ the polar coordinates are defined unambiguously for $\displaystyle 0<\theta<2\pi$. Find $\displaystyle \frac{\partial}{\partial{r}}$,$\displaystyle \frac{\partial}{\partial{\theta}}$. Find the relationship between these vectors and $\displaystyle \frac{\partial}{\partial{x}}|$,$\displaystyle \frac{\partial}{\partial{y}}|$.

1. By "common coordinates" I think you mean "Cartesian Coordinates"

2. I don't know what the "|" is in $\displaystyle \frac{\partial }{\partial x}|$ and $\displaystyle \frac{\partial }{\partial y}|$

3. If you mean the partial derivatives, use the chain rule: $\displaystyle \frac{\partial U}{\partial r}= \frac{\partial x}{\partial r}\frac{\partial U}{\partial x}+ \frac{\partial y}{\partial r}\frac{\partial U}{\partial y}$.
• May 7th 2010, 10:16 AM
Drexel28
Quote:

Originally Posted by HallsofIvy
1. By "common coordinates" I think you mean "Cartesian Coordinates"

2. I don't know what the "|" is in $\displaystyle \frac{\partial }{\partial x}|$ and $\displaystyle \frac{\partial }{\partial y}|$

3. If you mean the partial derivatives, use the chain rule: $\displaystyle \frac{\partial U}{\partial r}= \frac{\partial x}{\partial r}\frac{\partial U}{\partial x}+ \frac{\partial y}{\partial r}\frac{\partial U}{\partial y}$.

Isn't common coordinates a reference to overlap maps?